The result ends Argyle's six-year stay in the second tier, with Sheffield Wednesday and Crystal Palace's final-day encounter meaning it is impossible for them to stay up even if they win their remaining two games.
Paul Mariner's men have paid the price for winning just 11 games all season, while Newcastle notched their 29th victory of the campaign.
The hosts started brightly but failed to take some decent early chances before Newcastle found their game and Carroll headed home Barton's corner.
Routledge coolly slotted home Newcastle's second goal
Barton made their second with a defence-splitting pass to Routledge, the winger coolly rounding keeper David Stockdale and slotting into an empty net.
Plymouth showed passion and commitment in the second half, with Judge denied by the woodwork.
But substitute Leon Best almost made it 3-0 late on when his 25-yard shot also found the bar, and an instinctive save by Stockdale kept out a close-range Barton effort.
At the other end, defender Jose Enrique denied Plymouth a late consolation when he cleared Joe Mason's header off the line, but by then the Toon Army had already started partying.
Newcastle boss Chris Hughton:
"I think we got to a stage where we couldn't shout from the rooftops but we were comfortable that we were going to go up.
"We got to that stage and probably for a little while it's been about winning the title and I couldn't be any prouder of this group of players.
"They have gone from day one and we have probably had one blip during the season, they came back from that strongly and went on a very good run. We have been the team to beat all season."
"We have a magnificent support that have made every stadium that we have gone to a big occasion, including tonight."
Plymouth manager Paul Mariner: "We have a fantastic fan base and we owe it to them to do our best to bounce back up to this division - and that's what we will be aiming to do.
Mariner 'feels pain' of relegation
"Obviously I am so disappointed but I still have to say that the players tried to get the ball down, tried to pass and play the right way - even under the pressure they were under tonight, knowing that they had to win.
"That speaks volumes to me, and re-affirms what they are about. I asked for a response after the Watford defeat and they gave me that. We had chances to score tonight and our finishing has to improve and that's what we need to do."
90:00+3:33 The referee signals the end of the game.
90:00+2:35 Corner taken by Alan Judge from the right by-line, Joe Mason takes a shot. Danny Simpson makes a clearance.
89:55 Shola Ameobi takes a shot. Save made by David Stockdale.
89:14 The referee blows for offside against Yannick Bolasie. Free kick taken by Steve Harper.
87:10 Shot by Joe Mason from deep inside the six-yard box clear the bar.
86:07 Danny Guthrie takes a shot. Blocked by Richard Eckersley. Shot on goal by Leon Best from inside the box goes harmlessly over
85:20 Joey Barton takes a shot. Save by David Stockdale.
83:52 Free kick awarded for an unfair challenge on Richard Eckersley by Leon Best. David Stockdale takes the direct free kick.
83:33 Leon Best takes a shot.
82:57 Substitution (Newcastle) makes a substitution, with Tamas Kadar coming on for Fitz Hall.
82:38 The assistant referee flags for offside against Jamie Mackie. Steve Harper takes the indirect free kick.
80:02 David Stockdale takes the free kick. Free kick awarded for an unfair challenge on Alan Smith by Reda Johnson. Direct free
kick taken by Steve Harper.
80:02 Substitution Yannick Bolasie is brought on as a substitute for Onismor Bhasera.
80:02 Wayne Routledge is penalised for a handball.
77:37 Joey Barton concedes a free kick for a foul on Alan Judge. Free kick taken by David Stockdale.
76:39 Joe Mason is caught offside. Steve Harper restarts play with the free kick.
74:41 Alan Judge has an effort at goal from outside the penalty box which goes wide left of the goal.
73:45 David Stockdale takes the free kick.
73:45 Substitution Joe Mason joins the action as a substitute, replacing Bradley Wright-Phillips.
73:45 David Stockdale fouled by Shola Ameobi, the ref awards a free kick.
71:54 Inswinging corner taken right-footed by Danny Guthrie, clearance made by Carl Fletcher. Free kick awarded for an unfair challenge
on Onismor Bhasera by Wayne Routledge. Direct free kick taken by Onismor Bhasera. Free kick awarded for a foul by Wayne Routledge
on Onismor Bhasera. Free kick taken by David Stockdale.
71:05 Unfair challenge on Alan Judge by Leon Best results in a free kick. David Stockdale restarts play with the free kick.
69:57 Substitution Andrew Carroll leaves the field to be replaced by Shola Ameobi.
69:57 Substitution Leon Best joins the action as a substitute, replacing Kevin Nolan.
69:31 The ball is delivered by Wayne Routledge, David Stockdale makes a save.
67:55 Foul by Danny Simpson on Jamie Mackie, free kick awarded. Alan Judge crosses the ball from the free kick right-footed from
left channel, save by Steve Harper.
66:20 Andrew Carroll takes a shot. Reda Johnson gets a block in.
64:10 Corner taken by Danny Guthrie, clearance made by Reda Johnson. Andrew Carroll concedes a free kick for a foul on Reda Johnson.
David Stockdale takes the free kick.
62:55 Long distance shot from Bradley Wright-Phillips misses to the left of the target.
59:07 Kevin Nolan gives away a free kick for an unfair challenge on Karl Duguid. David Stockdale restarts play with the free kick.
57:10 Effort from 30 yards by Alan Judge.
56:09 The offside flag is raised against Kevin Nolan. Carl Fletcher restarts play with the free kick.
54:49 Corner taken right-footed by Danny Guthrie from the left by-line to the near post, clearance made by Karl Duguid. Inswinging
corner taken right-footed by Danny Guthrie played to the near post, clearance by Bradley Wright-Phillips.
52:33 The offside flag is raised against Jamie Mackie. Free kick taken by Steve Harper.
51:35 Shot on goal by Kevin Nolan from just inside the area clears the crossbar.
50:29 Jamie Mackie produces a right-footed shot from long range which goes wide right of the goal.
49:12 Unfair challenge on Karl Duguid by Joey Barton results in a free kick. Reda Johnson takes the free kick.
48:29 Kevin Nolan takes a shot. Save by David Stockdale.
47:53 Foul by Jamie Mackie on Danny Simpson, free kick awarded. Mike Williamson takes the free kick.
47:28 Jamie Mackie takes a shot. Save made by Steve Harper.
45:01 The referee gets the second half started.
45:00+1:12 The referee blows for half time.
45:00+0:20 The official flags Wayne Routledge offside. David Stockdale takes the free kick.
43:49 Foul by Andrew Carroll on Carl Fletcher, free kick awarded. Direct free kick taken by David Stockdale.
43:03 Alan Judge produces a right-footed shot from just outside the box that goes wide right of the goal.
41:07 Shot from just outside the box by Bradley Wright-Phillips goes over the target.
40:07 Free kick awarded for an unfair challenge on Kevin Nolan by Reda Johnson. Direct free kick taken by Steve Harper.
35:05 The free kick is delivered right-footed by Alan Judge from left wing, Andrew Carroll manages to make a clearance.
35:05 Booking Andrew Carroll shown a yellow card.
34:54 Andrew Carroll concedes a free kick for a foul on Alan Judge.
29:50 Volleyed right-footed shot by Jamie Mackie. Steve Harper makes a save.
27:28 Assist by Joey Barton.
27:28 GOAL - Wayne Routledge:Plymouth 0 - 2 Newcastle Wayne Routledge fires in a goal from inside the six-yard box low into the middle of the goal. Plymouth 0-2 Newcastle.
26:03 Andrew Carroll challenges Reda Johnson unfairly and gives away a free kick. David Stockdale takes the free kick.
21:56 Jamie Mackie is caught offside. Steve Harper restarts play with the free kick.
19:46 Assist by Danny Guthrie.
19:46 GOAL - Andrew Carroll:Plymouth 0 - 1 Newcastle Andrew Carroll finds the net with a headed goal from close in. Plymouth 0-1 Newcastle.
19:26 Corner from the right by-line taken by Danny Guthrie,
18:24 Effort on goal by Andrew Carroll from deep inside the area goes harmlessly over the bar.
16:50 Free kick awarded for an unfair challenge on Joey Barton by Alan Judge. Steve Harper takes the direct free kick.
16:08 Foul by Danny Guthrie on Damien Johnson, free kick awarded. The free kick is swung in right-footed by Damien Johnson, Mike
Williamson makes a clearance.
13:37 Free kick awarded for an unfair challenge on Alan Smith by Carl Fletcher. Direct free kick taken by Sanchez Jose Enrique.
11:54 Foul by Danny Simpson on Jamie Mackie, free kick awarded. David Stockdale restarts play with the free kick.
7:12 Sanchez Jose Enrique fouled by Damien Johnson, the ref awards a free kick. Sanchez Jose Enrique takes the free kick.
6:02 Carl Fletcher has an effort at goal from 20 yards. Mike Williamson gets a block in.
4:07 Foul by Alan Smith on Carl Fletcher, free kick awarded. Direct free kick taken by David Stockdale.
2:17 Unfair challenge on Reda Johnson by Kevin Nolan results in a free kick. David Stockdale takes the free kick.
0:52 Jamie Mackie takes a shot. Steve Harper makes a save. Outswinging corner taken right-footed by Alan Judge from the right by-line,
Bondz N'Gala produces a header from inside the six-yard box that clears the bar.
0:00 The game begins.
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